On statistical convergence in fractal analysis

Jun-Jie Quan; Selim Çetin; Ömer Kişi; Mehmet Gürdal; Qing-Bo Cai; Jun-Jie Quan; Selim Çetin; Ömer Kişi; Mehmet Gürdal; Qing-Bo Cai

doi:10.3934/math.2025812

AIMS Mathematics

2025, Volume 10, Issue 8: 18197-18215. doi: 10.3934/math.2025812

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Research article Special Issues

On statistical convergence in fractal analysis

1.
Department of Mathematics and Digital Sciences, Chengyi College, Jimei University, Xiamen 361021, China
2.
Department of Mathematics, Burdur Mehmet Akif Ersoy University, Burdur, Turkey
3.
Department of Mathematics, Bartin University, Bartın, Turkey
4.
Department of Mathematics, Süleyman Demirel University, 32260, Isparta, Turkey
5.
Fujian Provincial Key Laboratory of Data-Intensive Computing, Key Laboratory of Intelligent Computing and Information Processing, School of Mathematics and Computer Science, Quanzhou Normal University, Quanzhou 362000, China

Received: 23 May 2025 Revised: 18 July 2025 Accepted: 04 August 2025 Published: 12 August 2025
MSC : 28A80, 31E05, 40A05, 40G05

This study investigated the statistical convergence of fractal-generating set sequences, motivated by the observation that natural fractals, influenced by external biological, chemical, or physical factors, rarely exhibit strict classical convergence. Instead, their limiting behavior often aligns with statistical patterns. We formalized the concept of statistical convergence for compact subsets of $ \mathbb{R}^n $, introduced the notion of statistical Cauchy sequences, and established their sufficiency for statistical convergence—mirroring the classical relationship. Several illustrative examples and graphical simulations, including variants of the Sierpiński triangle and Koch snowflake, highlight the distinction between classical and statistical convergence. The proposed framework provides a more realistic and robust approach to understanding fractal structures in both theoretical and applied contexts.
- Hausdorff distance,
- fractal analysis,
- statistical convergence of sets,
- statistical Cauchy sequences,
- Banach fixed point theorem,
- box dimension
Citation: Jun-Jie Quan, Selim Çetin, Ömer Kişi, Mehmet Gürdal, Qing-Bo Cai. On statistical convergence in fractal analysis[J]. AIMS Mathematics, 2025, 10(8): 18197-18215. doi: 10.3934/math.2025812

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Abstract

This study investigated the statistical convergence of fractal-generating set sequences, motivated by the observation that natural fractals, influenced by external biological, chemical, or physical factors, rarely exhibit strict classical convergence. Instead, their limiting behavior often aligns with statistical patterns. We formalized the concept of statistical convergence for compact subsets of $ \mathbb{R}^n $, introduced the notion of statistical Cauchy sequences, and established their sufficiency for statistical convergence—mirroring the classical relationship. Several illustrative examples and graphical simulations, including variants of the Sierpiński triangle and Koch snowflake, highlight the distinction between classical and statistical convergence. The proposed framework provides a more realistic and robust approach to understanding fractal structures in both theoretical and applied contexts.

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